The problem is to find the equation of the hyperplane tangent to the ellipse $\displaystyle x_1^2+ 2x_2^2+ 3x_3^2+ \cdot\cdot\cdot+ nx_n^2= \frac{n(n+1)}{2}$ at the point $\displaystyle (-1, -1, -1, \cdot\cdot\cdot, -1)$.
Now, what is your difficulty? Can you find the gradient of the function? That will give a vector perpendicular to the ellipse at the give point. Do you know how to write the equation of a hyperplane given a normal vector and point?